• Nenhum resultado encontrado

1.2 Lógica Para Computação: Uma Introdução Curta -

1.2.7 Conclusões e Perspectivas

O estudo de sistemas lógicos têm sido fundamental desde as origens da Ciência da Computação, a ponto da lógica ser citada como o cálculo da computação [25]. Recen-temente, temos presenciado resultados muito promissores de pesquisas em Ciência da Computação e Inteligência Artificial [43, 50], que demonstram que os sistemas com-putacionais atuais já são capazes de integrar habilidades cognitivas complexas, como o aprendizado a partir de experiências e o raciocínio sobre o conhecimento adquirido do ambiente. A construção de sistemas computacionais com complexidade crescente, exige, portanto, conhecimento sofisticado de técnicas, modelos e fundamentos da Ciên-cia da Computação.

Os grandes desafios científicos sempre exigiram e, cada vez mais, irão demandar habilidade e capacidade de formalizar e construir raciocínios não triviais em sistemas de computação. Este capítulo ofereceu uma introdução mínima e incompleta à lógica aplicada. Esperamos que o leitor, a partir da bibliografia, possa explorar novos caminhos de forma independente. Para concluir de forma promissora, citamos Michael Rabin, vencedor do ACM Turing Award em 1976:

Our field is still in its embryonic stage. It’s great that we haven’t been around for 2000 years. We are at a stage where very, very important results occur in front of our eyes.M.O. Rabin em [44]

Referências

[1] Dirk Baltzly. Stoicism. In Edward N. Zalta, editor,The Stanford Encyclopedia of Philosophy. Spring 2014 edition, 2014.

[2] Rafael V. Borges, Artur d’Avila Garcez, and Luís C. Lamb. Learning and repre-senting temporal knowledge in recurrent networks. IEEE Transactions on Neural Networks, 22(12):2409–2421, 2011.

[3] K. Broda, S. Eisenbach, H. Khoshnevisan, and S. Vickers. Reasoned Program-ming. Prentice Hall, 1994.

[4] K. Broda, D.M. Gabbay, L.C. Lamb, and A. Russo. Compiled Labelled Deduc-tive Systems: A Uniform Presentation of Non-classical Logics. Studies in Lo-gic and Computation. Research Studies Press/Institute of Physics Publishing, Bal-dock, UK, Philadelphia, PA, 2004.

[5] Alonzo Church. An unsolvable problem of elementary number theory. American Journal of Mathematics, 58(2):345–363, April 1936.

[6] Stephen A. Cook. The complexity of theorem-proving procedures. InProceedings of the 3rd Annual ACM Symposium on Theory of Computing, May 3-5, 1971, Sha-ker Heights, Ohio, USA, pages 151–158, 1971.

[7] Flavio Corrêa da Silva, Marcelo Finger, and Ana C. V. de Melo. Lógica para Computação. Thomson, São Paulo, 2006.

[8] Artur d’Avila Garcez, Marco Gori, Pascal Hitzler, and Luís C. Lamb. Neural-Symbolic Learning and Reasoning (Dagstuhl Seminar 14381). Dagstuhl Reports, 4(9):50–84, 2015.

[9] A.S. d’Avila Garcez and L.C. Lamb. A connectionist computational model for epistemic and temporal reasoning. Neural Computation, 18(7):1711–1738, 2006.

[10] A.S. d’Avila Garcez, L.C. Lamb, and D.M. Gabbay. Connectionist computations of intuitionistic reasoning. Theoretical Computer Science, 358(1):34–55, 2006.

[11] A.S. d’Avila Garcez, L.C. Lamb, and D.M. Gabbay. Connectionist modal lo-gic: Representing modalities in neural networks. Theoretical Computer Science, 371(1-2):34–53, 2007.

[12] A.S. d’Avila Garcez, L.C Lamb, and D.M. Gabbay. Neural-Symbolic Cognitive Reasoning. Cognitive Technologies. Springer, 2009.

[13] Leonardo M. de Moura and Nikolaj Bjørner. Satisfiability modulo theories: intro-duction and applications. Commun. ACM, 54(9):69-77, 2011.

[14] Leo de Penning, Artur S. d’Avila Garcez, Luís C. Lamb, and John-Jules Ch.

Meyer. A neural-symbolic cognitive agent for online learning and reasoning. In Toby Walsh, editor,IJCAI-11, pages 1653–1658. IJCAI/AAAI, 2011.

[15] R. Fagin, J. Halpern, Y. Moses, and M. Vardi. Reasoning about Knowledge. MIT Press, 1995.

[16] Frederic B. Fitch. Symbolic Logic. The Ronald Press Company, New York, 1952.

[17] M. Fitting.Proof methods for modal and intuitionistic logics. D. Reidel Publishing Company, Dordrecht, 1983.

[18] D. Gabbay, A. Kurucz, F.Wolter, and M. Zakharyaschev.Many-dimensional modal logics: theory and applications, volume 148 ofStudies in Logic and the Founda-tions of Mathematics. Elsevier Science, 2003.

[19] D. M. Gabbay. Elementary Logics: a Procedural Perspective. Prentice Hall, London, 1998.

[20] Dov M. Gabbay and F. Guenthner, editors. Handbook of Philosophical Logic, volume I-XVIII. Springer, 2008-2015.

[21] Dov M. Gabbay, Amir Pnueli, Saharon Shelah, and Jonathan Stavi. On the tem-poral basis of fairness. InConference Record of the Seventh Annual ACM Sympo-sium on Principles of Programming Languages, Las Vegas, Nevada, USA, January 1980, pages 163–173, 1980.

[22] Dov M. Gabbay and John Woods, editors. Handbook of The History of Logic, volume I-XI. Elsevier, 2008-2015.

[23] G. Gentzen. Unstersuchungen über das logische Schliessen. Mathematische Zeits-chrift, 39:176–210, 1934.

[24] J.Y. Halpern. Reasoning about Uncertainty, MIT Press, 2003.

[25] J.Y. Halpern, R. Harper, N. Immerman, P.G. Kolaitis, M.Y. Vardi, and V. Vianu.

On the unusual effectiveness of logic in computer science. Bulletin of Symbolic Logic, 7(2):213–236, 2001.

[26] J. Hintikka. Knowledge and Belief. Cornell University Press, 1962.

[27] C. A. R. Hoare. Communicating Sequential Processes. Prentice-Hall, 1985.

[28] C. Howson. Probability and logic. J. Applied Logic1(3-4), 151-165, 2003.

[29] M. Huth and M. Ryan. Logic in Computer Science: modelling and reasoning about systems. Cambridge University Press, 2000.

[30] R. Khardon and D. Roth. Learning to reason. Journal of the ACM, 44(5):697–725, 1997.

[31] Robert A. Kowalski. Computational Logic and Human Thinking: How to be Arti-ficially Intelligent. Cambridge University Press, 2011.

[32] Robert A. Kowalski. Logic for Problem Solving, Revisited. Herstellung und Ver-lag: Books on Demand, 2014.

[33] S. Kripke. A completeness theorem in modal logic. Journal of Symbolic Logic, 24:1–4, 1959.

[34] S. Kripke. Semantic analysis of modal logics I, normal propositional calculi. Zeits-chrift für mathematische Logik und Glundlagen der Mathematik, 9:67–96, 1963.

[35] C. Lewis. A Survey of Symbolic Logic. University of California Press, Berkeley, 1918.

[36] Z. Manna and R. Waldinger.The logical basis for computer programming. Volume 1: deductive reasoning. Addison-Wesley, Boston, 1985.

[37] E. Mendelson. Introduction to mathematical logic. Van Nostrand Reinhold, New York, 1964.

[38] Robin Milner. Communication and concurrency. PHI Series in computer science.

Prentice Hall, 1989.

[39] Joan Moschovakis. Intuitionistic logic. In Edward N. Zalta, editor,The Stanford Encyclopedia of Philosophy. Spring 2015 edition, 2015.

[40] A. Pnueli. The temporal logic of programs. InProceedings of 18th IEEE Annual Symposium on Foundations of Computer Science, pages 46–57, 1977.

[41] S.J. Russell. Unifying Logic and Probability: A New Dawn for AI? InInformation Processing and Management of Uncertainty in Knowledge-Based Systems - 15th International Conference, IPMU 2014, pages 10–14, 2014.

[42] Ralph Schoenman, editor. Bertrand Russell: Philosopher of the Century. Allen and Unwin, London, 1967.

[43] Bernhard Schoelkopf. Artificial intelligence: Learning to see and act. Nature, 518:486–487, 2015.

[44] Denis Shasha and Cathy Lazere. Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists. Copernicus, 1995.

[45] R. M. Smullyan. First-Order Logic. Dover Publications, New York, revised edi-tion, 1995.

[46] Alan M. Turing. On computable numbers, with an application to the Entschei-dungsproblem. Proceedings of the London Mathematical Society, 42(2), 1936.

[47] Alan M. Turing. Computing machinery and intelligence.Mind, 59:433–460, 1950.

[48] L. G. Valiant. Three problems in computer science.Journal of ACM, 50(1):96–99, 2003.

[49] Dirk Van Dalen. Intuitionistic logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 166 of Synthese Library, pages 225–

339. Springer Netherlands, 1986.

[50] Moshe Y. Vardi. Is information technology destroying the middle class? Commun.

ACM, 58(2):5, 2015.

[51] M.Y. Vardi. Why is modal logic so robustly decidable? In N. Immerman and P. Kolaitis, editors,Descriptive Complexity and Finite Models, volume 31 of Dis-crete Mathematics and Theoretical Computer Science, pages 149–184. DIMACS, 1997.

[52] Alfred North Whitehead and Bertrand Russell. Principia Mathematica, volume I,II, III. Cambridge University Press, Cambridge, 1910, 1912, 1913.

[53] Ludwig Wittgenstein. Tractatus Logico-Philosophicus. Kegan Paul, Trench, Tub-ner & Co, London, 1922.